The rate of convergence of dykstra's cyclic projections algorithm: The polyhedral case

Suppose K is the intersection of a finite number of closed half-spaces in a Hilbert space X. Starting with any point xεX, it is shown that the sequence of iterates {x n } generated by Dykstra's cyclic projections algorithm satisfies the inequality for all n, where P K (x) is the nearest point i...

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Vydané v:	Numerical functional analysis and optimization Ročník 15; číslo 5-6; s. 537 - 565
Hlavní autori:	Deutsch, Frank, Hundal, Hein
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Marcel Dekker, Inc 01.01.1994
Predmet:	alternating projections best approximations from polyhedra convex regression cyclic projections Hildreth's algorithm isotone regression iterative projections linear inequalities linear programming Primary 41A65 quadratic programming rate of convergence of Dykstra's algorithm Secondary 47N10 Secondary 49M30 subject classification
ISSN:	0163-0563, 1532-2467
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Popis
Shrnutí:	Suppose K is the intersection of a finite number of closed half-spaces in a Hilbert space X. Starting with any point xεX, it is shown that the sequence of iterates {x n } generated by Dykstra's cyclic projections algorithm satisfies the inequality for all n, where P K (x) is the nearest point in K to x;, ρ is a constant, and 0 ≤c<1. In the case when K is the intersection of just two closed half-spaces, a stronger result is established: the sequence of iterates is either finite or satisfies for all n, where c is the cosine of the angle between the two functionals which define the half-spaces. Moreover, the constant c is the best possible. Applications are made to isotone and convex regression, and linear and quadratic programming.
ISSN:	0163-0563 1532-2467
DOI:	10.1080/01630569408816580